We consider the problem of computing the square root of a perturbation of the scaled identity matrix, A = αIn + U V * , where U and V are n × k matrices with k ≤ n. This problem arises in various applications, including computer vision and optimization methods for machine learning. We derive a new formula for the pth root of A that involves a weighted sum of powers of the pth root of the k × k matrix αI k + V * U . This formula is particularly attractive for the square root, since the sum has just one term when p = 2. We also derive a new class of Newton iterations for computing the square root that exploit the low-rank structure. We test these new methods on random matrices and on positive definite matrices arising in applications. Numerical experiments show that the new approaches can yield much smaller residual than existing alternatives and can be significantly faster when the perturbation U V * has low rank.
Existing algorithms for computing the matrix cosine are tightly coupled to a specific precision of floating-point arithmetic for optimal efficiency so they do not conveniently extend to an arbitrary precision environment. We develop an algorithm for computing the matrix cosine that takes the unit roundoff of the working precision as input, and so works in an arbitrary precision. The algorithm employs a Taylor approximation with scaling and recovering and it can be used with a Schur decomposition or in a decomposition-free manner. We also derive a framework for computing the Fréchet derivative, construct an efficient evaluation scheme for computing the cosine and its Fréchet derivative simultaneously in arbitrary precision, and show how this scheme can be extended to compute the matrix sine, cosine, and their Fréchet derivatives all together. Numerical experiments show that the new algorithms behave in a forward stable way over a wide range of precisions. The transformation-free version of the algorithm for computing the cosine is competitive in accuracy with the state-of-the-art algorithms in double precision and surpasses existing alternatives in both speed and accuracy in working precisions higher than double.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.